Topological singularities arising from fractional-gradient energies

We prove that, on a planar regular domain, suitably scaled functionals of Ginzburg-Landau type, given by the sum of quadratic fractional Sobolev seminorms and a penalization term vanishing on the unitary sphere, $\mathrm{\Gamma }$-converge to vortex-type energies with respect to the flat convergence of Jacobians. The compactness and the $\mathrm{\Gamma }$-$\mathrm{lim\hspace{0.17em}inf}$ follow by comparison with standard Ginzburg-Landau functionals depending on Riesz potentials. The $\mathrm{\Gamma }$-$\mathrm{lim\hspace{0.17em}sup}$, instead, is achieved via a direct argument by joining a finite number of vortex-like functions suitably truncated around the singularity.